Difference between Adjacent strong edge coloring and vertex distinguishing strong edge coloring

What is the exact difference between the adjacent strong edge coloring and vertex distinguishing proper edge coloring of graphs? This paper refers to the fact that the adjacent strong edge coloring of all graphs without $ K_2$ or $ C_5$ component is less than $ \Delta+2$ where $ \Delta$ is the maximum degree of the graph as an open conjecture, but, whereas this paper gives a proof that the vertex distinguishing proper edge coloring is less than $ \Delta+2$ . But, the definitions seem to coincide. Am I missing something here? Thanks beforehand.